10n^2-20n=12

Solution for 10n^2-20n=12 equation:

Simplifying
10n2 + -20n = 12

Reorder the terms:
-20n + 10n2 = 12

Solving
-20n + 10n2 = 12

Solving for variable 'n'.

Reorder the terms:
-12 + -20n + 10n2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + -20n + 10n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-6 + -10n + 5n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-6 + -10n + 5n2)' equal to zero and attempt to solve:

Simplifying
-6 + -10n + 5n2 = 0

Solving
-6 + -10n + 5n2 = 0

Begin completing the square.  Divide all terms by
5 the coefficient of the squared term: 

Divide each side by '5'.
-1.2 + -2n + n2 = 0

Move the constant term to the right:

Add '1.2' to each side of the equation.
-1.2 + -2n + 1.2 + n2 = 0 + 1.2

Reorder the terms:
-1.2 + 1.2 + -2n + n2 = 0 + 1.2

Combine like terms: -1.2 + 1.2 = 0.0
0.0 + -2n + n2 = 0 + 1.2
-2n + n2 = 0 + 1.2

Combine like terms: 0 + 1.2 = 1.2
-2n + n2 = 1.2

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 1.2 + 1

Reorder the terms:
1 + -2n + n2 = 1.2 + 1

Combine like terms: 1.2 + 1 = 2.2
1 + -2n + n2 = 2.2

Factor a perfect square on the left side:
(n + -1)(n + -1) = 2.2

Calculate the square root of the right side: 1.483239697

Break this problem into two subproblems by setting 
(n + -1) equal to 1.483239697 and -1.483239697.

Subproblem 1
n + -1 = 1.483239697

Simplifying
n + -1 = 1.483239697

Reorder the terms:
-1 + n = 1.483239697

Solving
-1 + n = 1.483239697

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '1' to each side of the equation.
-1 + 1 + n = 1.483239697 + 1

Combine like terms: -1 + 1 = 0
0 + n = 1.483239697 + 1
n = 1.483239697 + 1

Combine like terms: 1.483239697 + 1 = 2.483239697
n = 2.483239697

Simplifying
n = 2.483239697

Subproblem 2
n + -1 = -1.483239697

Simplifying
n + -1 = -1.483239697

Reorder the terms:
-1 + n = -1.483239697

Solving
-1 + n = -1.483239697

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '1' to each side of the equation.
-1 + 1 + n = -1.483239697 + 1

Combine like terms: -1 + 1 = 0
0 + n = -1.483239697 + 1
n = -1.483239697 + 1

Combine like terms: -1.483239697 + 1 = -0.483239697
n = -0.483239697

Simplifying
n = -0.483239697

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.483239697, -0.483239697}
Solution
n = {2.483239697, -0.483239697}